Page:EB1911 - Volume 22.djvu/390

as an outrigger and prevents the vessel heeling over. The main hull carries the mast rigging and an enormous triangular-shaped sail.

PROBABILISM (from Lat. probare, to test, approve), a term used both in theology and in philosophy with the general implication that in the absence of certainty probability is the best criterion. Thus it is applied in connexion with casuistry for the view that the layman in difficult matters of conscience may safely follow a doctrine inculcated by a recognized doctor of the church. This view was originated by the monk Molina (1528-1581), and has been widely employed by the Jesuits. In philosophy the term is applied to that practical doctrine which gives assistance in ordinary matters to one who is sceptical in respect of the possibility of real knowledge: it supposes that though knowledge is impossible a man may rely on strong beliefs in practical affairs. This view was held by the sceptics of the New Academy (see Scepticism and Carneades). Opposed to “probabilist” is “probabiliorism” (Lat. probabilior, more likely), which holds that when there is a preponderance of evidence on one side of a controversy that side is presumably right.

PROBABILITY (Lat. probabilis, probable or credible), a term which in general implies credibility short of certainty.

The mathematical theory of probabilities deals with certain phenomena which are employed to measure credibility. This Description and Division of the Subject. measurement is well exemplified by games of chance. If a pack of cards is shuffled and a card dealt, the probability that the card will belong to a particular suit is measured by—we may say, is—the ratio 1:4, or 1/4; there being four suits to any one of them the card might have belonged. So the probability that an ace will be drawn is 4/52, as out of the 52 cards in the pack 4 are aces. So the probability that ace will turn up when a die is thrown is 1/6. The probability that one or other of the two events, ace or deuce, will occur is 1/3. If simultaneously a die is thrown and a card is dealt from a pack which has been shuffled, the probability that the double event will consist of two aces is 1 × 4 divided by 6 × 52, as the total number of double events formed by combining a face of a die with a face of a card is 6 × 52, and out of these 1 × 4 consist of two aces.

The data of probabilities are often prima facie at least of a type different from that which has been described. For example, the probability that a child about to be born will be a boy is about 0.51. This statement is founded solely on the observed fact that about 51% of children born (alive, in European countries) prove to be boys. The probability is not, as in the instance of dice and cards, measured by the proportion between a number of cases favourable to the event and a total number of possible cases. Those instances indeed also admit of the measurement based on observed frequency. Thus the number of times that a die turns up ace is found by observation to be about $16.{\dot {6}}\%$ of the number of throws; and similar statements are true of cards and coins.^[1] The probabilities with which the calculus deals admit generally of being measured by the number of times that the event is found by experience to occur, in proportion to the number of times that it might possibly occur.

The idea of a probable or expected number is not confined to the number of times that an event occurs; if the occurrence of the event is associated with a certain amount of money or any other measurable article there will be a probable or expected amount of that article. For instance, if a person throwing dice is to receive two shillings every time that six turns up, he may expect in a hundred throws to win about $2\times 16.{\dot {6}}$ (about $33.{\dot {3}}$) shillings. If he is to receive two shillings for every six and one shilling for every ace, his expectation will be $2\times 16.{\dot {6}}+1\times 16.{\dot {6}}$ (50) shillings. The expectation of lifetime is calculated on this principle. Of 1000 males aged ten say the probable number who will die in their next year is 490, in the following year 397, and so on; if we (roughly) estimate that those who die in the first year will have enjoyed one year of life after ten, those who die in the next year will have enjoyed two years of life, and so on; then the total number of years which the 1000 males^[2] aged ten may be expected to live is

1 × 1000 + 2 × (1000 - 490) + 3 (1000 - 490 - 397) + . . .

Space as well as time may be the subject of expectation. If drops of rain fall in the long run with equal frequency on one point—or rather on one small interval, say of a centimetre or two—on a band of finite length and negligible breadth, the distance which is to be expected between a point of impact in the upper half of the line and a point of impact in the lower half has a definite proportion to the length of the given line.^[3]

Expectation in the general sense may be considered as a kind of average.^[4] The doctrine of averages and of the deviations therefrom technically called “errors” is distinguished from the other portion of the calculus by the peculiar difficulty of its method. The paths struck out by Laplace and Gauss have hardly yet been completed and made quite secure. The doctrine is also distinguished by the importance of its applications. The theory of errors enables the physicist so to combine discrepant observations as to obtain the best measurement. It may abridge the labour of the statistician by the use of samples.^[5] It may assist the statistician in testing the validity of inductions.^[6] It promises to be of special service to him in perfecting the logical method of concomitant variations; especially in investigating the laws of heredity. For instance the correlation between the height of parents and that of children is such that if we take a number of men all of the same height and observe the average height of their adult sons, the deviation of the latter average from the general average of adult males bears a definite proportion—about a half—to the similarly measured deviation of the height common to the fathers. The same kind and amount of correlation between parents and children with respect to many other attributes besides stature has been ascertained by Professor Karl Pearson and his collaborators.^[7] The (kinetics of free molecules (gases) forms another important branch of science which involves the theory of errors.

The description of the subject which has been given will explain the division which it is proposed to adopt. In Part I. probability and expectation will be considered apart from the peculiar difficulties incident to errors or deviation from averages. The first section of the first part will be devoted to a preliminary inquiry into the evidence of the primary data and axioms of the science. Freed from philosophical difficulties the mathematical calculation of probabilities will proceed in the second section. The analogous calculation of expectation will follow in the third section. The contents of the first three sections will be illustrated in the fourth by a class of examples dealing with space measurements—the so-called “local” or “geometrical” probabilities. Part II. is devoted to averages and the deviations therefrom, or more generally that grouping of statistics which may be called a law of frequency. Part II. is divided into two sections distinguished by differences in character and extent between the principal generalizations respecting laws of frequency.

Part I.—Probability and Expectation

Section I.—First Principles.

1. As in other mathematical sciences, so in probabilities, or even more so, the philosophical foundations are less clear than the calculations based thereon. On this obscure and controversial topic absolute uniformity is not to be expected. But it is hoped that the following summary in which diverse authoritative judgments are balanced may minimize dissent.

2. (1) How the Measure of Probability is Ascertained.—The first question which arises under this head is: on what evidence are the facts obtained which are employed to measure probability? A very generally accepted view is that which Laplace has thus expressed:

1. Cf. note to par. 5 below.
2. It is more usual to speak of the mean expectation, the average number of years per head.
3. Below, par. 88.
4. For more exact definition see below, par. 95.
5. See Bowley's Address to Section F. of the British Association (1906).
6. Edgeworth, “Methods of Statistics,” Journal of the Statistical Society (Jubilee volume, 1885).
7. See Biometrika, vol. iii. “Inheritance of Mental Characters.”